NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
Assessment of a Simply Supported Plate with Uniformly Distributed Load
A building has a floor opening that has been covered by a durbar plate with a yield stress of 275MPa.
The owner has been instructed by his insurers that for safety the load carrying capacity of the plate
needs to be assessed. The owner has calculated (possibly unrealistically but certainly conservatively)
that if 120 people each weighing 100kg squeeze onto the plate then it must be able to cope with
100kN/m2. He has found, in the Steel Designers’ Manual, that the plate should be able to withstand
103kN/m2. This is rather close to the required load and looking in Roark’s Formulas for Stress and
Strain he finds that the collapse load is more like 211kN/m2 which he feels does provide an adequate
factor of safety. However, with the huge difference between the two published values he has asked
you to provide him with an independent assessment of the load carrying capacity of the plate.
Figure 1: Symmetric quarter model of challenge plate and boundary conditions
Raison d’être for the Challenge This challenge derives from an observation that the strength of steel plates, such as the one in this
challenge, as quoted in the Steel Designers’ Manual is significantly lower than that derived from
standard reference texts such as Roark. The strength of a steel plate such as the challenge plate can
be assessed using limit analysis techniques. Ramsay Maunder Associates (RMA) has developed a
software tool for the lower-bound limit analysis of steel plates. This software tool uses equilibrium
finite element (EFE) technology and will be used to assess the strength of the challenge plate. As
specialist limit analysis software will not be available to most practising engineers the problem will
need to be tackled using a simulated limit analysis with incremental non-linear material finite
elements.
The Challenge
As an experienced engineer you realise that under increasing load the plate will eventually reach first yield after which the stress will
redistribute until the final collapse load is reached. You will appreciate that the steel will have some work hardening capability and that if
transverse displacements are considered then some membrane action will occur. However, opting for simplicity and realising that
ignoring these two strength enhancing phenomena will lead to a degree of conservatism in your assessment, you decide that this is a limit
analysis problem in which the flexural strength of the plate governs collapse.
Unless you have specialist limit analysis software you will decide to tackle this as an incremental non-linear plastic problem with a bi-
linear stress/strain curve and a von Mises yield criterion.
Please carry out an assessment of the strength of the plate and provide your best estimate of the actual collapse load together with
evidence of the verification you have conducted sufficient to convince the owner and his risk averse insurer.
Plate Properties
Length (�) = 2m
Breadth (�) = 0.6m
Thickness (�) = 0.01m (10mm)
Simply supported with an applied pressure (�)
� � 210��
� 0.3
X
Y
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
Review of Published Strength Values
Corrigendum – The Owner Bodged his Calculation! Sharp readers will have noted that the owner of the building made a mistake using the equation
from Roark. He calculated �� correctly but then assumed that this was the applied pressure or UDL.
However, this value is the total load and so the pressure is obtained by dividing the total load by the
area of the plate:
� ���
�����7.68 ∙ 275�� ∙ 0.01�
2 ∙ 0.6 = 176���
It is interesting to notice that whilst the collapse pressure is dependent on the area of the plate, the
collapse load is not.
Plate Theory and Boundary Conditions Classical plate theory recognises the need for different formulations for ‘thick’ and ‘thin’ plates. For
‘thick’ plates the appropriate theory is Reisner-Mindlin, which accounts for shear deformation,
whereas for ‘thin’ plates the appropriate theory is Kirchhoff which assumes the section to be rigid in
terms of shear deformation. The transition between ‘thick’ and ‘thin’ plates occurs at a span to
depth ratio of about ten and as the challenge problem has a ratio of 0.6/0.01=60 it is clearly in the
realm of ‘thin’ plate or Kirchhoff theory. In theory, at least, the results produced by ‘thick’ plate
theory should converge to those achieved by ‘thin’ plate theory as the span to depth ratio increases.
However, one needs to be aware that with many finite element formulations of ‘thick’ plates, a
nasty phenomenon known as ‘shear-locking’ may occur to stymy this principal. Fortunately for the
challenge problem the span to depth ratio is not sufficiently large that shear-locking will occur and
finite elements of either formulation can be used successfully to solve this problem.
Values without an asterisk cause deflection greater than b/100
at serviceability, assuming that the only dead load present is
due to self-weight.
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
The formulation used for a plate element has implications on the nature of the boundary conditions
that might be applied. The thick formulation allows torsional moments and their corresponding
twisting rotations to be specified at the boundary whereas the thin formulation does not explicitly
control these quantities. As such, for a thick plate, the support conditions may be specified as ‘soft’
or ‘hard’ depending on whether the twisting rotations are free or constrained. As the challenge
problem is a thin plate problem this distinction is not relevant BUT if the challenge plate is modelled
with a thick plate finite element formulation then the simple supports should be interpreted as soft
simple supports and the twisting rotation left free.
As with any problem that exhibits symmetry it is worth taking advantage of this property in order to
reduce the size of the finite element model and therefore the time to compute a solution. For the
challenge problem the upper right hand quarter of the plate will be modelled as shown in figure 1.
The figure shows the boundary conditions, (soft) simply supports on the external edges and
symmetry conditions (no normal rotation) on the symmetry boundaries. The applied pressure is
indicated by a central arrow.
Elastic Solution A good starting point for understanding the challenge problem is to look at the elastic solution. The
finite element software used to generate the elastic solutions was CS1 and the results for a load of
100kPa are shown in figure 2.
Figure 2: Convergence of stress and displacement for an elastic analysis of the challenge plate
The maximum displacement and stresses occur at the centre of the plate and
this figure shows the convergence of these quantities with mesh refinement for
both four-noded and eight-noded elements. The values have been normalised
by the values achieved with the most refined eight-noded mesh:
Svm = 232MPa; Syy = 262MPa; dz = 8.5mm
The normalised peak von Mises stress and peak Syy stress are virtually identical
and so the curves lie on top of each other in the figure.
It is interesting to note that the direction in which convergence occurs is
dependent on the element type with results converging from above the true
solution for the eight-noded element and below the true solution for the four-
noded element.
There is an often presented argument (see [1] for example) that as
displacement elements are over-stiff then under applied forces they will tend to
deform less and therefore induce less stress than would occur in the actual
member. This is of course true in an integral sense and this can be seen in the
strain energy convergence for a pure displacement element model under
applied forces. In terms of point as opposed to integral values of quantities it is
possible however for individual points to behave differently. It is also the case
that some elements used in commercial FE systems are not pure displacement
elements and so for a number of reasons these arguments are spurious and may
not generally hold as seen with this example.
No
rma
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d R
esp
on
se
Degrees of Freedom
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
From the elastic analysis we have sufficient results to make the following statements:
1) The peak von Mises stress is 232MPa for a load of 100kPa so that first yield is reached at
275/232=1.185 times the applied pressure which is 118.5kPa.
2) Even if no redistribution occurs across the face of the plate, the applied pressure to cause
plastic collapse is 1.5x118.5=178kPa – where the 1.5 factor comes from the increase in load
required to develop yield from first yield at the surface to full yield through the section.
It is curious that this value is a little greater than the Roark value of 176kPa. So the Roark value is
clearly too conservative especially, as can be seen in the appendix, redistribution across the plate
does occur!
The Steel Designers’ Manual (SDM) use Pounder’s equations which are reproduced from his paper
[2] below:
Max Skin Stress - Pounder equation 19
34������ �1 + 1475 1 − �" + 2057 1 − �"�#
Maximum Deflection - Pounder equation 19(a)
� − 1 � ∙ 5���$32��% ∙ �1 + 37175 1 − �" + 79201 1 − �"�#
where:
� = �$�$ + �$
Pounder’s equations are based on an elastic analysis and for a pressure of 100kPa applied to the
challenge plate give a maximum direct stress of 268MPa (which is pleasingly close to the finite
element value of 262MPa) and a maximum displacement of 8.72mm (again close to the finite
element value of 8.5mm). Comparing the maximum stress with the yield stress gives 275/268=1.03
which corresponds to an applied pressure of 103kPa and is identical to the value reported in the
SDM. The approach used by the SDM then is one which limits the maximum principal stress to the
yield stress for the material. Thus whilst the SDM quotes these loads as ‘Ultimate Load Capacity’ the
method allows no plastic redistribution and whilst appropriate for brittle materials, is not relevant
for the sort of ductile plates being considered – see figure 9 for practical evidence of ductility. In
terms of deflection, then the deflection at ‘failure’ is 1.03x8.72=9mm. It is interesting to note that in
the SDM, and based on a breadth/100=6mm deflection limit, the displacement is greater than this
value so the asterisk indicating that SLS is not a concern is incorrect!
The SDM suggests that Pounder’s equations take into consideration the phenomenon of corner
uplift. However, after several readings of Pounder’s paper, phrases indicating consideration of
corner uplift were not found and so it should be assumed that this phenomenon is not considered.
This is not surprising since corner uplift is a non-linear phenomenon that is not simply dealt with in
the form of linear analysis considered in Pounder’s paper. To consider corner uplift one can perform
an incremental finite element analysis progressively releasing any boundary nodes where the
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
reactions are tensile. Such an analysis has been performed with linear elastic material properties
and although not reported here, the change in the maximum stress and displacement is observed to
be minimal.
Limit Analysis Solution The idea of limit analysis is to find the collapse load of the structure based on plasticity arguments
and a perfectly-plastic material model. For plates such as the challenge plate it is the flexural failure
mode that dominates. The collapse solution, in limit analysis, is found directly without recourse to
an incremental form of analysis. The yield line technique is perhaps the most widely known form of
limit analysis. In this technique the engineer postulates a (kinematically admissible) collapse
mechanism comprising sagging and hogging yield lines and then calculates the corresponding
collapse load. In general if the collapse mechanism is not the correct one then the technique
produces an upper-bound (unsafe) approximation to the true collapse load. To improve yield line
solutions some form of geometric optimisation is generally conducted on the collapse mechanism.
The collapse mechanism for the quarter model of the challenge slab comprises two sagging yield
lines and is simply described with a single geometric variable, ', as shown in figure 3.
Figure 3: Geometric optimisation of a yield line collapse mechanism
The optimised collapse load by yield line comes out at about 216kPa with a value of ' = 0.55(. It
should be recorded that the collapse load as predicted by the yield line technique is unlikely to be
exact since the technique adopts the Nielsen yield criterion rather than the von Mises criterion that
should be used for ductile steels – see [3].
RMA develop engineering software based on equilibrium finite elements that produce safe lower-
bound limit analysis approximations to the collapse load of structural members such as steel plates
using the von Mises yield criterion. This software Equilibrium Finite Elements or EFE has been used
to produce a reference solution for the challenge problem. The solution achieved was 231kPa and a
contour plot of material utilisation (yield moment divided by the von Mises moment) is shown in
'
Minimum at 216kPa
Co
lla
pse
Pre
ssu
re i
n k
Pa
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
figure 4. This result is verified by two of the challenge responders both of whom modelled the plate
with solid elements. The results from one of the responders are reproduced in the appendix.
Note: EFE uses triangular elements (see figure 8) and so the plate has been discretised into regular rectangular regions
which are then subdivided into four triangular elements. The convergence of collapse pressure with mesh is shown in the
table where the safe lower-bound characteristic can be observed with collapse loads converging rapidly from below the
true value.
Figure 4: Utilisation contours from EFE for a quarter of the challenge plate
The plot of utilisation shows that, with the exception of a small region around the centre of the long
supported edge, the material is fully utilised. The pressure to produce first yield has already been
calculated as 118.5kPa which leads (without stress redistribution across the surface of the plate) to a
conservative prediction of collapse at 178kPa. The value achieved by EFE, together with the plot of
utilisation, shows that significant stress redistribution does occur across the surface of the plate and
the evolution of this redistribution can be seen in the appendix.
It is interesting to compare the collapse pressures produced by EFE with those tabulated in Roark.
Table 1 reproduces Roark’s results and is appended with results achieved by EFE and the percentage
difference.
b/a 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 ) (Roark) 5.48 5.50 5.58 5.64 5.89 6.15 6.70 7.68 9.69 ) (EFE) 6.26 6.28 6.40 6.51 6.90 7.42 8.38 10.08 13.67
Difference 14% 41%
Table 1: Roark’s values for * appended with those produced by EFE
It is interesting to see that using dedicated limit analysis software more capacity can be achieved
from one’s structure – the results from EFE are greater than those produced by Roark. In the text,
Roark admits that the results are not expected to be accurate and that they could be up to 30% in
error. In practice whilst Roark’s values are conservative, for the plate aspect ratios considered, they
can be greater than 40% in error!
Simulated Limit Analysis Solutions Bespoke limit analysis software is only recently becoming available for the practising engineer and in
the absence of such tools engineers wishing to undertake this challenge have used simulated limit
analysis through incremental non-linear material finite element analysis. Whereas bespoke limit
analysis tools require minimal input data and user intervention, conventional finite element systems
used to predict the collapse of plates will require more careful consideration if reliable results are to
be achieved. To this end a Tips and Hints document on simulating limit analysis results was
produced [4].
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
The following figure is reproduced from [4] and shows how the moment/rotation develops with
increasing load at a point in a beam or plate.
Figure 5: Moment/Rotation development through the thickness of a plate
If a problem can be concocted where the moment field is constant over the plate then the plot in
figure 5 will apply to all points in the plate and the only stress redistribution that can occur will be
through the thickness of the plate. Such a constant moment problem was presented in [4] and is
reproduced in figure 6.
Figure 6: A constant moment plate problem
The manner in which the bending moments redistribute is governed by the chosen yield criterion.
The von Mises yield criterion, which is appropriate to steel members and thus for the challenge
problem, is shown in figure 7 where it is presented in terms of normalised principal moments which
are simply the principal moments ((+and (�) divided by the collapse moment:
(, = -.��4 = 6875Nm/m
x
y
2 (3(.(3.4 = 5 6��07
Geometric Properties: � = 1(; � = 0.01(
Elastic Properties: � = 210��; : = 0.3
Plastic Properties: -. = 275;��
�<(/(
�<(/(
The yellow line in this
figure indicates how
the stress at the
surface increases
linearly until it reaches
yield. The green line
indicates the degree of
penetration of yield
through the section.
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
Plastic collapse is governed by the yield criterion and the von Mises yield criterion may be written in
terms of the principal moments:
(+� + (�� − (+(� ≤ (,�
Figure 7: The von Mises yield criterion in terms of normalised principal moments
The load factor, 5, is a non-dimensional constant that scales the applied loads and we are interested
in determining the value at which the plate will collapse. As the moment field for this problem
involves no torsional moments then the principal moments are simply equal to the normal Cartesian
components (+ = 5� and (� = 5�. We can thus determine the load factor at collapse,5,, in terms
of the collapse moment and the amplitudes of the unfactored normal moments:
5, = (,√�� + �� − ��
With unit values for the direct moment components, the load factor to cause collapse is equal to the
amplitude of the collapse moment:
� = � = 1 → 5, = 6875
The constant moment problem just described is ideal for verifying a finite element system because it
has a known exact solution which should be reproduced using a single finite element. This problem
will be analysed using the popular commercial finite element tools CS1 and CS2.
As already discussed, a finite element system could get away with a single plate element of the
‘thick’ type provided the formulation of the element was robust. But, as evidenced by the large
The figure shows two von Mises
ellipses the smaller one being that of
first yield and the large one being that
of collapse moment. The inset figures
in the corners shows the principal
moments acting on an infinitesimal
element of the plate (green and blue
for the first and second principal
moments respectively). They also
show the direction of the direct
stresses at the top surface of the plate
developed by these applied moments.
The red arrow traces the principal
moments for the constant moment
problem.
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
number of plate/shell type elements offered in many commercial systems, this ideal seems not to
have been realised. This situation is not helpful to the practising engineer who is then faced with
selecting an element for his/her plate problem from a wide range of available elements.
A second issue in the simulation of limit analysis with conventional finite element systems is that the
solution one is trying to detect is the point at which the structure loses its stiffness when the
deflections and strains tend to infinity. This means that one needs to wait for the programme to
abort and then have the necessary faith to believe the answer!
The following table presents the finite element collapse moments for the constant moment
problem. For both commercial FE systems four noded shell elements with reduced integration were
used. In CS1 the element is described as ‘a four-node finite strain shell’ whilst for CS2 the element is
described as ‘a four-node doubly curved thin or thick shell, reduced integration, hourglass control,
finite membrane strains’. Default solution parameters were used and the number of integration
points through the plate thickness was varied. The load applied was 10,000Nm per edge so that the
time at collapse for the exact collapse load should be 0.6875 in a 1 second time frame.
Number Integration Points Through Thickness CS1 CS2
3 / 4583.8 (33.3%)
5 (Default) 6506.3 (5.4%) 6875.4 (0%)
7 6562.5 (4.5%) 6620.4 (3.7%)
9 (Correct Solution) 6875.0 (0%) 6875.1 (0%)
11 / 6783.2 (1.3%)
Table 2: Collapse moments (Nm/m) for two commercial FE systems (constant moment problem)
The default integration scheme for both FE codes is five integration points through the plate
thickness. The results indicate, for CS1, that this integration scheme is not sufficient to recover the
true solution. It seems, however, that a nine point scheme can provide accurate recovery of the true
solution for both CS1 and CS2 although it is interesting that, for CS2, a further increase in the
number of integration points leads to a less accurate solution.
Collapse pressures for the challenge plate were generated using fairly coarse meshes in the two
commercial finite element systems. The experience gained from the constant moment problems in
terms of integration points through the thickness was utilised with nine points being used. The total
pressure applied was 1000kPa so that the time at collapse should be around 0.231 in a 1 second
time frame. Like the constant moment problem, default solution settings were used for both
commercial software tools.
CS1 CS2
Mesh Reduced Full Reduced Full
1x3 387 385 1000 + 386
3x10 235 235 1000 + 241
6x20 225 224 435 234
Table 3: Collapse pressures (kPa) for two commercial FE systems (challenge problem)
The first point to note with regard to the performance of the commercial FE codes on the challenge
problem is that convergence is from above the true value. This is important since, at least for the
challenge problem, coarse meshes produce unsafe predictions of the collapse load c.f. the results
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
from EFE which converge from below the true value and are thus always safe irrespective of mesh
refinement – see figure 4. It appears also to be the case that reduced integration schemes have a
detrimental effect on the prediction of the collapse pressure. This is particularly significant for CS2
which would appear to show that the entire pressure can be taken for the coarser meshes with the
reduced integration element. Despite the above concerns, it is pleasing to see, at least for the fully
integrated elements, that the collapse load is predicted fairly reasonably (at least in an engineering
sense) by both codes. The convergence of the collapse pressure for the challenge plate is shown in
figure 8.
Figure 8: Convergence of the collapse pressure for the challenge plate
Discussion This challenge has proved interesting at a number of levels. In developing the limit analysis tool EFE
and testing it on published results, RMA has been long aware that the results presented in the SDM
and in Roark, whilst conservative, are incorrect.
The results published in the SDM are based on the elastic equations of Pounder which, although
stated as taking account of corner uplift do not include this non-linear phenomenon – and anyway
corner uplift does not significantly influence the maximum deflection and stress for such a plate
configuration. The SDM uses Pounder’s equations to predict the maximum principal stress in the
plate and then factors this with the yield stress to provide what it calls the ‘Ultimate Load Capacity’.
This phrase is generally understood to mean the load at which failure will occur and in the context of
steel plates one would generally interpret this as the load at which failure by collapse occurs.
However, it is clear, from this study, that the meaning implied in the SDM is the load at which first
yield occurs. This might of course mean actual failure if the plate were made of a brittle material
Number of Elements
No
rma
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ap
se P
ress
ure
(No
rma
lisi
ng
Pre
ssu
re =
23
1k
Pa
)
Unsafe Region
Safe Region
CS1, CS2 (Reduced) and CS2 lie virtually on top of each other.
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
(Pounder’s work did deal with cast steel plates) but as used by the SDM in the context of Durbar
plates, which offer considerable ductility – see figure 9, this is rather misleading.
In contrast to the SDM, Roark makes clear that it is talking about the plastic collapse of plates so we
understand the loads it provides to be collapse loads. The results are produced in tabular format for
a practical range of plate aspect ratios. The result for the challenge plate aspect ratio is interesting
since it is less than 1.5 times the load to produce first yield. It is known, even without allowing for
any plastic redistribution across the plate, that the redistribution through the thickness will give a
collapse pressure 1.5 times that required to produce first yield and so, as in general redistribution
across the plate can and does occur (see appendix), it is clear that this result is going to be rather
conservative. The accompanying text in Roark states that the results are likely to be up to 30% in
error (hopefully on the safe side - although not specified!) and it is seen using the dedicated and
verified limit analysis software EFE that for the largest aspect ratio plate considered in the table, the
error is actually greater than 40% - see table 1.
The published results for the collapse load of plates such as the challenge plate are thus rather
disappointing and do not serve the practising engineer who might be designing a new plate or
assessing an existing plate for a change in duty.
The yield line analysis of the plate gives a collapse load of 216kPa which is some 6% below the
correct (verified) value. This serves to remind us that whilst the yield line technique is appropriate
for the limit analysis of reinforced concrete slabs which adhere to the Nielsen yield criterion, when
used for metallic plates that obey the von Mises yield criterion differences may occur. In this case
the yield line result is conservative but there are other cases where it can be considerably unsafe.
Whilst RMA have the bespoke limit analysis software EFE to conduct collapse analyses on plates such
as the challenge plate, it was realised that most practising engineers would not have access to such
software and would need to use conventional finite element codes to simulate the limit analysis
solution. To assist in the understanding and verification of simulated limit analysis [4] was prepared.
A constant moment problem was presented in this document which possessed a known exact
solution both in terms of elasticity and plasticity and for which a finite element system should be
able to recover the exact solution with a single four-noded plate/shell element. Analysis of this
problem with two commercial FE systems showed that with the default settings one of the systems
was incapable of capturing the exact result. Further investigation showed that the default
integration scheme through the plate thickness was the cause for this issue and that increasing from
the default five points to nine points through the thickness cured this problem.
Simulated limit analysis using incremental procedures in a conventional finite element system was
recognised as a rather unsatisfactory procedure since one is attempting to predict, with accuracy, a
point of structural instability where the displacements and strains tend to infinity. A high degree of
reliance/faith on the solution settings is required as the solution only appears when the software has
aborted the solution process. Nonetheless, the default settings in the two commercial FE systems
tested for this challenge problem did seem appropriate and led to results for the challenge problem
that were not unreasonable. It was noted however that for the plate elements used, convergence
with mesh refinement was from above the exact solution. This form of convergence is potentially
unsafe since if the engineer does not perform adequate mesh refinement then the prediction will be
greater than the exact value. It was also noted for one of the commercial FE systems (CS2) that the
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
reduced integration shell element was incapable of converging on the true collapse load. It is not
surprising then that of the responders who used plate/shell elements to model the challenge plate
there was a considerable range of results submitted.
Two responders used solid elements to model the challenge plate. Both demonstrated convergence
to a solution which was virtually identical to that produced by EFE. The results from one of these
analyses are presented in the appendix and served to verify EFE.
Thinking from the perspective of the practising engineer then it has become clear that whilst
conventional FE systems can simulate limit analysis problems they are rather unsuited to the
problem. Different elements perform in different and sometimes unpredictable manners, the
default settings are sometimes not appropriate and the need to rely on a myriad of solution settings
for the software to abort at the correct load seems a rather unsatisfactory way of going about
things. As usual there is also the essential requirement for good simulation governance by way of
verification through mesh convergence studies and when this is not appropriately applied, at least
for the plate elements tested, the results might be considerably unsafe. In contrast to this a
dedicated or bespoke limit analysis software tool such as EFE requires no the setting of and reliance
on solution parameters and achieves a guaranteed safe solution, irrespective of mesh refinement in
a fraction of the time taken to simulate the problem in conventional FEA – even for the most refined
mesh the solution in EFE took less than a second whereas for the solid models used by some of the
respondents the solution took in excess of 5 minutes.
Closure Beyond simple linear elastic analysis, limit analysis provides further useful information regarding one
of the possible failure modes of a structure, i.e. plastic collapse. It provides invaluable information
about how much more load your structure can take beyond that predicted from a simple linear-
elastic analysis or, alternatively, it enables you to make the most of your available structural
capacity. It provides another tool for the engineer to use in the understanding of his/her structure
and its response to load.
Published collapse loads for plates aimed at helping the practising engineer are seen to be
significantly different (although conservative) from the result produced through limit analysis. This
means that engineers using these texts will design structures with more material than is really
necessary and/or they will underestimate the residual strength when reassigning the duty of a
structural plate element potentially unnecessarily requiring its strengthening or replacement.
Neither situation is satisfactory. Commercially available finite element software tools are able to
simulate limit analysis solutions with reasonable accuracy BUT the results they produce are strongly
dependent on the engineer being skilled in non-linear FEA and having the time and interest in
ensuring appropriate simulation governance or verification.
Limit analysis is conservative since it ignores strength enhancing phenomenon such as strain
hardening and membrane actions. Whilst large permanent deformations might, in some
applications, be considered unacceptable, there are plenty of examples of plates, such as the
challenge plate, where large permanent deformations do not affect the serviceability of the plate –
see figure 9 for example.
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
Figure 9: Permanent and large deflections of a plate
The challenge, as presented, was for engineers to ‘convince the owner of the building and his risk
averse insurer’ that the plate, as fitted by the owner, was fit for duty. Given that the collapse
pressure is 231kPa then this is some 2.3 times the maximum load the plate is ever likely to see.
The owner of the building (Ferdinand Frugal of Frugal Castle) was rather peeved at the outcome of
this study in that he realises he could have got away with using a thinner (8mm instead of 10mm)
plate:
�@AB > D100��� ∙ 10�231��� = 6.58((
The difference in the cost of the plate is some £64 which Ferdinand feels would have been better
spent on a case of cheap Whisky for his guests!
Figure 10: Cost of Durbar plate for the challenge problem
References [1] ‘The fib Model Code for Concrete Structures 2010: A Critique of the Comments on Finite Element
Analysis’ (http://www.ramsay-maunder.co.uk/r--d/resources/ )
[2] C.C. Pounder, ‘The Design of Flat Plates’, The Association of Engineering and Shipbuilding
Draughtsmen, 4th
Edition, 1949.
[3] ‘Yield Line Analysis of Reinforced Concrete Slabs: is the 10% Rule Safe?’ http://www.ramsay-
maunder.co.uk/downloads/bm_yield_line_article.pdf
[4] ‘Simulated Limit Analysis of Plates using Incremental Plasticity Finite Elements’
(http://www.ramsay-maunder.co.uk/r--d/resources/ )
NBR Number 02 (January 2015 – April 2015)
Copyright © Ramsay Maunder Associates Limited (2004 – 2015). All Rights Reserved
Appendix – Development of Plasticity and Verification of EFE The results shown in this appendix were generated by Matt Watkins of ESRD Inc. using his
company’s finite element software StressCheck. The software provides objective solution
verification information by p-refinement which is an essential component of simulation governance.
Like one other responder Matt used solid elements to model the challenge plate and the collapse
pressure from both responders who used solid elements produced collapse pressures of about
231kPa agreeing almost exactly with the result produced by EFE.
The following figure shows the development of utilisation with increasing pressure from the elastic
solution through to collapse. The development of plasticity across the plate is clearly seen with the
utilisation at collapse being rather similar to that produced by EFE – c.f. figure 4.
103kPa
118
(first yield)
142
170
231
(collapse)
Figure 11: Utilisation contours from the elastic solution to collapse